Note: this is step 3 of a print / sinter / polish process.
Now that you've successfully printed with "metal" filament, and possibly sintered a solid metal part, you want to make it look nice. These steps are generally chronological, but you can always go out-of-order or do a step again. If you do plan to sinter, I highly recommend a pass of trimming & shaping before the sinter, then further cleanup after.
Trimming: use side-cutters or a hobby knife to clean up any nibs, blobs, or stray filament. Half-round diamond files work well too -- they have a sharp edge, and the clay-like filament comes right up (smoothly).
Shaping: I really love cheap nail files for sanding & shaping. I also have a set of diamond files by the printer for very small work or reaming misshapen holes. "Tweakers" (kind of like dental tools) work really well, Harbor Freight usually has some for $5ish.
When sanding / shaping, a little friction creates a little heat, and can cause the material to flow a little (almost like clay). This will smooth out layer lines without removing so much material. This can be a real problem when using a rotary tool, but is very manageable when hand sanding.
Fine grit works really well: green (unsintered) material is very soft and easy to work. For sintered parts, start at 150 and/or lean on your rotary tool; bronze can be a lot more stubborn to reshape. For plastics or green parts, start around 220 or higher, to avoid hard-to-remove scratch marks on the parts.
Tumbling: a $40 rock tumbler works great to quickly smooth a lot of parts, and if left in for a VERY long time can have some interesting effects. Plastics can shine up nicely with steel pins, or ceramic balls. Solid bronze needs a lot more persuasion, so I use coarse ceramic triangles. In all cases some burnishing compound or Lemi-shine can help keep things clean while tumbling. Be sure to check the parts every few hours, so you can stop them when they reach the desired texture.
I use this custom rock tumbler chamber inside my actual tumbler; it helps a lot.
Aging / Verdigris: it's not too hard to make a new part look very, very old. Copper-based alloys (copper, brass, bronze) form a blue-green verdigris, while iron alloys will "rust" and form one of a few types of Iron Oxide (all of which are red).
Iron Oxide: brush or sand the part to expose the iron particles, then submerge it in hydrogen peroxide. That extra oxygen will react quickly and help redden the part. After further processing, clear rustoleum will help protect the rust, but can turn it brown (there's a further reaction involved). Mod podge will maintain more of the color, and is easier to use when thinned with water.
Verdigris is actually a few different compounds, but there are two easy-to-acheive types which look great: darker copper (II) acetate, or lighter copper carbonate (copper(II) carbonate hydroxide). The final appearance is pretty different; decide which you prefer.
Darker copper (II) acetate is made with a solution of salt in vinegar: mix in a lot of rock salt or kosher salt (until it stops dissolving), soak the parts in it, then leave them suspended above it in the vapor in the presence of oxygen (air). I made a little tray to help... You can also brush wet vinegar/salt and leave the parts in open air, but it may take longer. Clean parts will react more quickly, so a pass with a brush or sandpaper, followed by soap/water, may help.
Lighter copper carbonate works with ammonia, or even Brasso (which contains ammonia and baking soda / sodium bicarbonate). Brasso, left for just a few minutes, will start turning bright blue-green directly on the part. Unfortunately it also wipes back off easily, so it can be tricky to use. Straight up ammonia (like for housecleaning) can be brushed on and left to react in air. I have also had success with mixing brasso in water and brushing that onto the target parts.
In both cases the finish can be protected with a nonreactive, clear finish like carnauba wax or mod podge. For a permanent protection, try ProtectaClear.
Polishing: once the part is in the right shape (surface texture), you'll want to make it look like actual metal. Many polishing kits are available, and some harder to use than others. For polishing plastics, because they are so soft, you really need to use high-grit papers and polishing compounds (Mothers is a favorite). Basically you start low (300-ish grit) and work your way up, progressively making smaller "scratches" until you get the desired results.
The dremel polishing kit has two plastic flappy wheels that seem to do really well with metal-filled plastics; I think they help smash the top layer flat + smooth, and partly expose the metal. They will shine up very nicely without removing a lot of material.
For solid bronze parts, I start with pickling, then a rotary tool + wire brush, a stone wheel, then a felt wheel (with some rouge or polishing compound) on a dremel. Sandpaper at about 150 grit works well for shaping.
Buff with a jewelry cloth (same as for cleaning jewelry or silverware)
Finishing: after all that, and especially with any active processes (like rust or verdigris) a final finishing coat can help pause chemical reactions and maintain the surface appearance. Clear rustoleum works great, but reacts with the red iron oxide and turns it brown. Thinned mod podge is very nice. Carnauba wax (Minwax furniture wax) is also very good, especially for holding a shine on solid metal. For jewelry I like ProtectaClear -- it holds a shine and doesn't add any teture.
Cheating: My favorite shortcut here: trim, shape, and corrode (rust or verdigris) the entire part. Apply a coat of mod podge or rustoleum to protect the interior surfaces. Run the parts through a tumbler for a while with coarse media, to clean the exterior surfaces again (this also removes any mod podge or rustoleum) This leaves the inside surfaces "old" and the outsides "new". Then proceed with polishing (which can only hit those same outside surfaces), and apply wax when you're done. The parts will look "old" and weathered, but handled / maintained.